Nomenclature

• $1$, Inlet of the stator
• $2$, Exit of the stator
• $3$, Inlet of the rotor
• $4$, Exit of the rotor

Input parameters

Thermodynamic variables

• Working fluid name
• $p_{01}$, stagnation pressure at turbine inlet
• $h_{01}$, stagnation enthalpy at turbine inlet
• $p_{4}$, static pressure at turbine outlet
• $\dot{m}$, mass flow rate

Turbine design parameters

• $\nu= u_3 / v_0$, blade velocity ratio
• $R=(h_3-h_4)/(h_1 - h_4)$, degree of reaction
• $\alpha_1$, absolute flow angle at the inlet of the stator
• $\alpha_2$, absolute flow angle at the exit of the stator
• $r_1/r_2$, radius ratio across stator
• $r_2/r_3$, radius ratio across interspace
• $r_3/r_4$, radius ratio across rotor
• $AR_{\mathrm{S}}=\tfrac{1}{2}(H_1 + H_2) / (r_2- r_1)$, aspect ratio of the stator
• $AR_{\mathrm{R}}=\tfrac{1}{2}(H_3 + H_4) / (r_4- r_3)$, aspect ratio of the rotor
• $\xi_\mathrm{S}= (h_2 - h_{2s})/0.5v_2^2$, loss coefficient of the stator
• $\xi_\mathrm{R}= (h_4 - h_{4s})/0.5w_4^2$, loss coefficient of the rotor
• $Z_\mathrm{S}= 0.8$, Zweiffel parameter of the stator
• $Z_\mathrm{R}= 0.8$, Zweiffel parameter of the rotor

Outputs

The flow coefficient $\phi=v_{m}/u_3$ is computed from the stage equations

The flow angles $\beta_3$ and $\beta_4$ are computed from the stage equations

The spouting velocity is given by

$$v_0 = \sqrt{2(h_{01} - h_{4s})}$$

The blade velocity at each station is given by

$$\begin{gather} u_1 = 0 \\ u_2 = 0 \\ u_3 = (r_3/r_4) \\ u_4 = \nu \, v_0 \end{gather}$$

The constant meridional velocity is given by:

$$v_m = \phi \, u_4$$

The absolute velocities are given by:

$$\begin{gather} v_1 = v_m / \cos(\alpha_1) \\ v_2 = v_m / \cos(\alpha_2) \\ v_3 = v_m / \cos(\alpha_3) \\ v_4 = v_m / \cos(\alpha_4) \\ \end{gather}$$

The relative velocities are given by:

$$\begin{gather} w_1 = v_m / \cos(\beta_1) \\ w_2 = v_m / \cos(\beta_2) \\ w_3 = v_m / \cos(\beta_3) \\ w_4 = v_m / \cos(\beta_4) \\ \end{gather}$$

The enthalpy of each station is given as

$$h_1 = h_{01} - v_{1}^2/2$$

The enthalpy at the exit of the stator os obtained from the equation for the conservation of energy:

$$\frac{h_1 - h_2}{u_3^2} = \frac{1}{2} \phi^2 \left( \tan^2 \alpha_2 - \tan^2 \alpha_1 \right)$$

In the absence of losses and for constant meridional velocity, the enthalpy does not change across the interspace:

$$h_3 = h_2, \quad \alpha_3 = \alpha_2$$

The enthalpy at the exit of the rotor is obtained from the conservation of rothalpy:

$$\frac{h_3 - h_4}{u_3^2} = \frac{1}{2} \phi^2 \left( \tan^2 \beta_4 - \tan^2 \beta_3 \right)- \frac{1}{2}(1-\rho^2)$$

Once the enthalpy at each flow station is known, the thermodynamic state can be calculated assuming an isentropic process accross the turbine

$$\begin{gather} [p_1, \rho_1, a_1, q_1] = \mathrm{EOS}(h_1, s_1) \\ [p_2, \rho_2, a_2, q_2] = \mathrm{EOS}(h_2, s_2) \\ [p_3, \rho_3, a_3, q_3] = \mathrm{EOS}(h_3, s_3) \\ [p_4, \rho_4, a_4, q_4] = \mathrm{EOS}(h_4, s_4) \\ \end{gather}$$

The equations for conservation of mass are used in order to compute all the radii and blade heights at all stations. From the mass balance in the stator we find that:

$$\frac{H_2}{H_1} = \frac{\rho_1}{\rho_2} \, \frac{r_1}{r_2}$$

from the definition of aspect ratio of the stator

$$\frac{H_2}{r_2} = 2 AR_{\mathrm{S}}( 1 - r_1/r_2)(1 + H_1/H_2)^{-1}$$

Inserting these results into the definition of mass flow rate we find that

$$\begin{gather} \dot{m} = \rho_2 v_m A_2 = \rho_2 v_m 2\pi r_2 H_2 \\ \dot{m} = \rho_2 v_2 2\pi r_2^2 \left[AR_{\mathrm{S}} \, ( 1 - r_1/r_2)(1 + H_1/H_2)^{-1} \right] \\ r_2^2 = \frac{\dot{m}}{4\pi v_m \rho_2} AR_{\mathrm{S}}^{-1} \, ( 1 - r_1/r_2)^{-1}(1 + H_1/H_2) \end{gather}$$

Once $r_2$ is known, the radius at the other stations is given by:

$$\begin{gather} r_1 = r_2 \cdot (r_1 / r_2) \\ r_3 = r_2 \cdot (r_2 / r_3)^{-1} \\ r_4 = r_3 \cdot (r_3 / r_4)^{-1} \end{gather}$$

The blade height at each station is thus given by:

$$\begin{gather} H_1 = \dot{m} (2\pi r_1 \rho_1 v_m)^{-1} \\ H_2 = \dot{m} (2\pi r_2 \rho_2 v_m)^{-1} \\ H_3 = \dot{m} (2\pi r_3 \rho_3 v_m)^{-1} \\ H_4 = \dot{m} (2\pi r_4 \rho_4 v_m)^{-1} \\ \end{gather}$$

In order to compute the number of blades and openng of the stator blades we can use the Zweiffel criterion to determine the spacing-to-chord ratio

$$\begin{gather} \frac{s_{\mathrm{S}}}{r_2- r_1} = \frac{0.5 Z_{\mathrm{S}}}{\cos^{2}\!\left( \alpha_{\mathrm{2}} \right) \left( \tan \alpha_{\mathrm{1}} - \tan \alpha_{\mathrm{2}} \right)} \\ N_{\mathrm{S}} = \pi (r_1 + r_2) / s_{\mathrm{S}} \\ o_{\mathrm{S}} = s_{\mathrm{S}} \cos(\alpha_2 + \tfrac{1}{2}\tfrac{2\pi}{N_{\mathrm{S}}}) \end{gather}$$

Similarly, for the rotor we have that:

$$\begin{gather} \frac{s_{\mathrm{R}}}{r_4- r_3} = \frac{0.5 Z_{\mathrm{R}}}{\cos^{2}\!\left( \beta_{\mathrm{4}} \right) \left( \tan \beta_{\mathrm{3}} - \tan \beta_{\mathrm{4}} \right)} \\ N_{\mathrm{R}} = \pi (r_4 + r_3) / s_{\mathrm{R}} \\ o_{\mathrm{R}} = s_{\mathrm{R}} \cos(\beta_4 + \tfrac{1}{2}\tfrac{2\pi}{N_{\mathrm{R}}}) \end{gather}$$